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b224933 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 | import Mathlib
import OAI.MeasureTheory.FalconerCompletion.Recursion.MaskScale
namespace OAI
/-! Dyadic shells and prefix decompositions. -/
section
section
open Set MeasureTheory
open scoped ComplexConjugate RealInnerProductSpace
namespace GraphLiteralSupport
namespace Attempt4FullFar
open Filter Real
open scoped Topology
noncomputable def shellEffectiveScaleND (l : ℕ) (R σ : ℝ) : ℝ :=
R^σ*sourceMaskScaleND l R σ
lemma dyadic_half_sqND (q : ℝ) : ((2:ℝ)^(q/2))^2=(2:ℝ)^q := by
rw [← Real.rpow_natCast ((2:ℝ)^(q/2)) 2,← Real.rpow_mul (by norm_num : (0:ℝ)≤2)]
congr 1
norm_num
lemma shell_effective_scale_upperND {R σ q : ℝ} (l : ℕ) (hR : 0<R)
(hl : (2:ℝ)^l ≤ (2:ℝ)^(q/2)*R^σ) :
shellEffectiveScaleND l R σ ≤ max (R^σ) ((2:ℝ)^(q/2)*R^(-48*σ)) := by
unfold shellEffectiveScaleND sourceMaskScaleND
rw [mul_max_of_nonneg _ _ (Real.rpow_nonneg hR.le _),mul_one]
apply max_le_max_left
calc
_ ≤ R^σ*(((2:ℝ)^(q/2)*R^σ)*R^(-50*σ)) := by gcongr
_ = (2:ℝ)^(q/2)*(R^σ*R^σ*R^(-50*σ)) := by ring
_ = _ := by rw [← Real.rpow_add hR,← Real.rpow_add hR]; congr 2; ring
lemma shell_effective_scale_sqND {R σ q : ℝ} (l : ℕ) (hR : 0<R)
(hl : (2:ℝ)^l ≤ (2:ℝ)^(q/2)*R^σ) :
(shellEffectiveScaleND l R σ)^2 ≤ max (R^(2*σ)) ((2:ℝ)^q*R^(-96*σ)) := by
have hu := shell_effective_scale_upperND l hR hl
have hn : 0 ≤ shellEffectiveScaleND l R σ :=
mul_nonneg (Real.rpow_nonneg hR.le _) (sourceMaskScaleND_pos l R σ).le
have he₁ : (R^σ)^2=R^(2*σ) := by
rw [← Real.rpow_natCast (R^σ) 2,← Real.rpow_mul hR.le]
congr 1
norm_num
ring
have he₂ : ((2:ℝ)^(q/2)*R^(-48*σ))^2=(2:ℝ)^q*R^(-96*σ) := by
rw [mul_pow,dyadic_half_sqND,← Real.rpow_natCast (R^(-48*σ)) 2,← Real.rpow_mul hR.le]
congr 2
norm_num
ring
rcases le_total (R^σ) ((2:ℝ)^(q/2)*R^(-48*σ)) with h | h
· rw [max_eq_right h] at hu
exact (pow_le_pow_left₀ hn hu 2).trans (by rw [he₂]; exact le_max_right _ _)
· rw [max_eq_left h] at hu
exact (pow_le_pow_left₀ hn hu 2).trans (by rw [he₁]; exact le_max_left _ _)
lemma shell_scale_regimeND {R σ q c Λ : ℝ} (l : ℕ)
(hR : 1 ≤ R) (hσ : 0 ≤ σ) (hσ₁ : σ ≤ 1) (hc : 0<c)
(hl : (2:ℝ)^l ≤ (2:ℝ)^(q/2)*R^σ)
(hqlo : R^(20*σ) ≤ (2:ℝ)^q) (hqup : (2:ℝ)^q ≤ R^2)
(hΛ : c*(2:ℝ)^q ≤ Λ) :
1 ≤ shellEffectiveScaleND l R σ ∧ shellEffectiveScaleND l R σ ≤ R^2 ∧
c*R^(20*σ) ≤ Λ ∧ (shellEffectiveScaleND l R σ)^2/Λ ≤ c⁻¹*R^(-18*σ) ∧
(2:ℝ)^(-q)*(sourceMaskScaleND l R σ)^2 ≤ 1 := by
have hRp : 0<R := zero_lt_one.trans_le hR
have hQp : 0<(2:ℝ)^q := Real.rpow_pos_of_pos (by norm_num) _
have hΛp : 0<Λ := (mul_pos hc hQp).trans_le hΛ
have hRs : 1≤R^σ := Real.one_le_rpow hR hσ
have hu := shell_effective_scale_upperND l hRp hl
have hsquare := shell_effective_scale_sqND l hRp hl
have hhalf : (2:ℝ)^(q/2) ≤ R := by
nlinarith [dyadic_half_sqND q, Real.rpow_nonneg (by norm_num : (0:ℝ)≤2) (q/2)]
have hle (u : ℝ) (hu : u ≤ 0) : R^u ≤ 1 := by
simpa using Real.rpow_le_rpow_of_exponent_le hR hu
have hnorm : (shellEffectiveScaleND l R σ)^2 ≤ (2:ℝ)^q*R^(-18*σ) := by
apply hsquare.trans
apply max_le
· calc
R^(2*σ) = R^(20*σ)*R^(-18*σ) := by rw [← Real.rpow_add hRp]; congr 1; ring
_ ≤ _ := mul_le_mul_of_nonneg_right hqlo (Real.rpow_nonneg hRp.le _)
· exact mul_le_mul_of_nonneg_left
(Real.rpow_le_rpow_of_exponent_le hR (by linarith)) hQp.le
refine ⟨?_,?_,?_,?_,?_⟩
· exact one_le_mul_of_one_le_of_one_le hRs (sourceMaskScaleND_one_le l R σ)
· apply hu.trans
apply max_le
· simpa using Real.rpow_le_rpow_of_exponent_le hR (show σ ≤ (2:ℝ) by linarith)
· calc
_ ≤ (2:ℝ)^(q/2) := mul_le_of_le_one_right (by positivity) (hle (-48*σ) (by linarith))
_ ≤ R := hhalf
_ ≤ R^2 := by nlinarith
· exact (mul_le_mul_of_nonneg_left hqlo hc.le).trans hΛ
· calc
_ ≤ ((2:ℝ)^q*R^(-18*σ))/(c*(2:ℝ)^q) :=
div_le_div₀ (by positivity) hnorm (mul_pos hc hQp) hΛ
_ = _ := by field_simp
· have hQle : sourceMaskScaleND l R σ ≤ shellEffectiveScaleND l R σ := by
exact le_mul_of_one_le_left (sourceMaskScaleND_pos l R σ).le hRs
have hsq : (sourceMaskScaleND l R σ)^2 ≤ (2:ℝ)^q :=
(pow_le_pow_left₀ (sourceMaskScaleND_pos l R σ).le hQle 2).trans
(hnorm.trans (mul_le_of_le_one_right hQp.le (hle (-18*σ) (by linarith))))
rw [Real.rpow_neg (by norm_num : (0:ℝ)≤2)]
exact (inv_mul_le_one₀ hQp).mpr hsq
theorem shell_scale_regime_eventuallyND {σ c : ℝ} (hσ : 0<σ) (hσ₁ : σ≤1) (hc : 0<c) :
∀ᶠ R : ℝ in atTop, ∀ (l : ℕ) (q Λ : ℝ),
(2:ℝ)^l ≤ (2:ℝ)^(q/2)*R^σ → R^(20*σ) ≤ (2:ℝ)^q → (2:ℝ)^q ≤ R^2 →
c*(2:ℝ)^q ≤ Λ →
1 ≤ shellEffectiveScaleND l R σ ∧ shellEffectiveScaleND l R σ ≤ R^2 ∧
R^σ ≤ Λ ∧ (shellEffectiveScaleND l R σ)^2/Λ ≤ R^(-σ) ∧
(2:ℝ)^(-q)*(sourceMaskScaleND l R σ)^2 ≤ 1 := by
filter_upwards [eventually_ge_atTop (1:ℝ),
(tendsto_rpow_atTop (by linarith : (0:ℝ)<17*σ)).eventually_ge_atTop c⁻¹] with R hR hlarge
intro l q Λ hl hqlo hqup hΛ
have hRp : 0<R := zero_lt_one.trans_le hR
obtain ⟨h₁,h₂,h₃,h₄,h₅⟩ := shell_scale_regimeND l hR hσ.le hσ₁ hc hl hqlo hqup hΛ
refine ⟨h₁,h₂,?_,?_,h₅⟩
· have hcR : 1 ≤ c*R^(17*σ) := by simpa only [mul_comm] using (inv_le_iff_one_le_mul₀ hc).mp hlarge
have hRs : R^σ ≤ R^(3*σ) := Real.rpow_le_rpow_of_exponent_le hR (by linarith)
apply le_trans _ h₃
calc
_ ≤ R^(3*σ) := hRs
_ ≤ (c*R^(17*σ))*R^(3*σ) := le_mul_of_one_le_left (Real.rpow_nonneg hRp.le _) hcR
_ = _ := by rw [mul_assoc,← Real.rpow_add hRp]; congr 2; ring
· apply h₄.trans
calc
_ ≤ R^(17*σ)*R^(-18*σ) := mul_le_mul_of_nonneg_right hlarge (Real.rpow_nonneg hRp.le _)
_ = _ := by rw [← Real.rpow_add hRp]; congr 1; ring
lemma shell_source_exponentsND {N l : ℕ} {σ q : ℝ}
(hl : (l:ℝ) ≤ q/2+σ*N) (hqlo : 20*σ*N ≤ q) (hqup : q ≤ 2*N) :
(2:ℝ)^l ≤ (2:ℝ)^(q/2)*((2:ℝ)^N)^σ ∧
((2:ℝ)^N)^(20*σ) ≤ (2:ℝ)^q ∧ (2:ℝ)^q ≤ ((2:ℝ)^N)^2 := by
have h₁ : ((2:ℝ)^N)^σ=(2:ℝ)^((N:ℝ)*σ) := by
rw [← Real.rpow_natCast (2:ℝ) N,← Real.rpow_mul (by norm_num : (0:ℝ)≤2)]
have h₂ : ((2:ℝ)^N)^(20*σ)=(2:ℝ)^((N:ℝ)*(20*σ)) := by
rw [← Real.rpow_natCast (2:ℝ) N,← Real.rpow_mul (by norm_num : (0:ℝ)≤2)]
have h₃ : ((2:ℝ)^N)^2=(2:ℝ)^((N:ℝ)*2) := by
rw [← Real.rpow_natCast (2:ℝ) N,← Real.rpow_natCast ((2:ℝ)^(N:ℝ)) 2,
← Real.rpow_mul (by norm_num : (0:ℝ)≤2)]
norm_num
rw [h₁,h₂,h₃,← Real.rpow_add (by norm_num : (0:ℝ)<2),← Real.rpow_natCast (2:ℝ) l]
exact ⟨Real.rpow_le_rpow_of_exponent_le (by norm_num) (by nlinarith),
Real.rpow_le_rpow_of_exponent_le (by norm_num) (by nlinarith),
Real.rpow_le_rpow_of_exponent_le (by norm_num) (by nlinarith)⟩
end Attempt4FullFar
end GraphLiteralSupport
end
end
end OAI
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